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Chapter 12: Problem 22

Graph. (GRAPH CANNOT COPY) $$y=-x-1$$

Short Answer

Expert verified
The graph of \(y = -x - 1\) will intercept the y-axis at point (0, -1) and will slope downwards to the right direction one unit for every unit moved to the right on the x-axis. Connect all the plotted points and extend the line on both ends to sketch the complete graph.

Step by step solution

01

Identifying the type of function

First of all, recognize that this equation represents a linear function. Linear functions follow the format \(y = ax + b\), where 'a' is the coefficient of x (which defines the slope), and 'b' is called the y-intercept (where the line intercepts the y-axis). In this case, our line has the equation \(y = -x - 1\), where 'a' (slope) is -1 and 'b' (y-intercept) is -1.
02

Determining the slope and y-intercept

So we have the slope and the y-intercept already. It is given by \(a = -1\) and \(b = -1\). This tells us two things: Because the slope \(a\) is -1, the graph of the function will slope downwards for every unit we move to the right along the x-axis. The y-intercept \(b\) is -1, which means the line intercepts the y-axis at point (0, -1).
03

Plotting the function

To accurately sketch the graph, begin at the y-intercept, which is at point (0, -1) on the graph. Because our slope is -1, for every one unit you move to the right on the x-axis, move one unit down on the y-axis. This will allow you to mark another point on the line. Once you have a few points on the line, connect them with a straight line through all points, extending the line past the points on both ends.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing
In the world of mathematics, graphing functions visually represents the relationship between variables. Linear functions are among the simplest to graph due to their straight-line nature. Plotting a linear function involves understanding its equation, which is usually in the form of \(y = ax + b\). Here, you're looking at a simple equation like \(y = -x - 1\).

To graph this function effectively:
  • Start by identifying the y-intercept, which is where the line crosses the y-axis. For this equation, it's the point (0, -1).
  • Next, use the slope to determine the direction and angle of the line. With a slope of -1, for every step to the right along the x-axis, move one step downward.
  • Plot this initial point and apply the slope to find additional points.
  • Once several points are marked, draw a straight line through them. This line can extend indefinitely in both directions.
Visualizing this method can help you quickly grasp the behavior of linear functions and apply it to any similar equation.
Slope
The slope is a fascinating attribute of linear functions. It tells you how steep the line is and the direction it goes. In a standard linear equation, \(y = ax + b\), the coefficient \('a'\) represents the slope.

In our specific equation, \(y = -x - 1\), we have:
  • A slope of -1, meaning the line descends at a 45-degree angle as you move right along the x-axis.
  • The negative sign indicates a downward slope, meaning that as x increases, y decreases.
  • Slope can also be considered as "rise over run." For every one unit you move horizontally (x-axis), the altitude (y-axis) changes by the slope's magnitude.
This concept of slope is universal across linear functions, assisting in predicting how all points lie relative to one another.
Y-Intercept
The y-intercept is a key feature in linear equations, indicating where the line crosses the y-axis. In the equation \(y = ax + b\), \('b'\) is your y-intercept. It's the value of y when x is 0.

For our equation of interest, \(y = -x - 1\):
  • The y-intercept is at \(b = -1\), meaning the line crosses the y-axis at the coordinate (0, -1).
  • This point is your starting reference when graphing. Knowing the y-intercept allows a quick grasp of the line's vertical position.
  • It鈥檚 an anchor point that facilitates finding the location of the line in the coordinate space.
Understanding the y-intercept helps provide a foundation for plotting the entire linear graph accurately.

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Most popular questions from this chapter

Determine whether the line through \(P_{1}\) and \(P_{2}\) is parallel, perpendicular, or neither parallel nor perpendicular to the line through \(Q_{1}\) and \(Q_{2}\). $$P_{1}(-2,4), P_{2}(2,4) ; Q_{1}(-3,6), Q_{2}(4,6)$$

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